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Advanced Computer Graphics. Spring 2002 Professor Brogan. Simulated Annealing. Monte Carlo approach for minimizing multivariate functions Monte Carlo = Random = Stochastic Requires one ‘goodness’ metric – result of evaluation function

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## Advanced Computer Graphics

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**Advanced Computer Graphics**Spring 2002 Professor Brogan**Simulated Annealing**• Monte Carlo approach for minimizing multivariate functions • Monte Carlo = Random = Stochastic • Requires one ‘goodness’ metric – result of evaluation function • Multivariate – selects multiple parameter values to minimize evaluation function**Algorithm Outline**• Select some initial guess of evaluation function parameters: x0 • Evaluate evaluation function, E(x0)=v • Compute a random displacement, x’0 • The Monte Carlo event • Evaluate E(x’0) = v’ • If v’ < v; set new state, x1 = x’0 • Else set x1 = x’0 with Prob(E,T) • This is the Metropolis step • Repeat with updated state and temp**What is Annealing?**• Used to treat work-hardened parts made out of low-carbon steels • Heat to a specific temperature, then soak, and then cool slowly • Thermodynamics – molecules can move around when they are at high temps. Slow cooling permits self organization into minimum energy configurations**Back to our Situation**• We approximate nature’s alignment of molecules by allowing uphill transitions • exp (-E/kT) Boltzmann Probabilty Distribution • Even when T is small, a disruption is possible • exp (-(E2-E1) / kT) Metropolis Step • The rate at which T is decreased and the amount it is decreased is prescribed by an annealing schedule**What have we got?**• Always move downhill if possible • Sometimes go uphill • Optimality guaranteed with slow annealing schedule • No need for smooth search space • No derivatives • Can be discrete search space • Traveling salesman problem**Optimization**• Given: • What value of minimizes f()? • Expensive to computer f()? • Expensive to compute partials of f()=Jacobian? • Global vs. local solutions**Constrained Optimization**• Minimize • Subject to: • There are a priori limitations on the possible values of the independent variables**Linear Programming**• Special type of constrained optimization • Function f() and constraints are linear

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